Max-semigroups of bivariate random variables with Khinchine-type decompositions (Q2754980)

Let \(S \subset R^2\) be a semigroup w.r.t. coordinate-wise maximum, i.e. \(\max(x,y) \in S\) when \(x\in S\) and \(y\in S\). To \(S\) corresponds the semigroup \(M(S)\), with pointwise multiplication, of distribution functions of \(S\)-valued random vectors: when \(X\) and \(Y\) are \(S\)-valued independent random vectors, so is \(Z=\max(x,y)\) and \(P(Z\leq u)=P(X\leq u)P(Y\leq u)\). The unit of \(S\) is its smallest element \(c\), when present, and then \(\delta(c,.)\) is the unit of \(M(S)\). NEWLINENEWLINENEWLINEThe paper derives conditions on \(S\) under which any \(F \in M(S)\) has a decomposition \(F=H\prod_iG_i\), analogous to Khinchin's decomposition for ordinary convolution, see \textit{Yu. V. Linnik} and \textit{I. V. Ostrovskij} [``Decomposition of random variables and vectors'' (1972; Zbl 0285.60009)]. The \(G_i\) should be irreducible, i.e. when \(G_i=VW\), then \(V=G_i\) or \(W=G_i\) and \(H\) anti-irreducible, i.e. reducible and when \(H=VW\) with \(W\) irreducible, then \(V=H\). It is proved that for closed \(S\), with or without unit, there is a decomposition (finite or infinite). For the difficult case of non-closed \(S\) with unit necessary and sufficient conditions are derived when \(S\) is a finite union of convex sets.